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Modulational instability for a logarithmic nonlinear Schrödinger equation
AlgebraLetters and its48Applications 466 (2015) 102–116 Applied Linear Mathematics (2015) 124–127
Contents lists at ScienceDirect Contents lists available at available ScienceDirect
LinearMathematics Algebra andLetters its Applications Applied www.elsevier.com/locate/laa www.elsevier.com/locate/aml
eigenvalue problem of Jacobi matrix Modulational Inverse instability for a logarithmic nonlinear Schr¨odinger with mixed data equation Takuya Yamano Ying Wei 1 Department of Mathematics and Physics, Faculty of Nanjing Science,University Kanagawa of University, 2946, Tsuchiya, Hiratsuka, Department of Mathematics, Aeronautics and6-233 Astronautics, Kanagawa 259-1293, Japan Nanjing 210016, PR China
article
i n faor t i c l e
ianbfs ot r a c t
a b s t r a c t
Article history: Article history: In this paper, the inverse problem of reconstructing We investigate the modulational instability for eigenvalue a logarithmic nonlinear Schr¨ odinger 2014 Received 20 March 2015Received 16 January equation a Jacobi matrix infrom eigenvalues, its leading principal that has recently proposed the its context of a wave packet dynamics Accepted 20 September 2014 Received in revised form 30 March submatrix and part process. of the eigenvalues of itslimit submatrix under a continuous quantum measurement In the long wave of the Available online 22 October 2014 2015 considered. Therate necessary and sufficient conditions for perturbation, we showis that the growth of the instability coincides with the Submitted by Y. Wei Accepted 31 March 2015 the existence and uniqueness of the solution are derived. friction coefficient contained in the equation. Available online 20 April 2015 Furthermore, a numerical algorithm and © 2015 Elsevier Ltd. Allsome rightsnumerical reserved. MSC: examples are given. 15A18 Keywords: © 2014 Published by Elsevier Inc. Modulational instability15A57 Logarithmic nonlinear Schr¨ odinger Keywords: equation Jacobi matrix Plane wave Eigenvalue Inverse problem Submatrix
1. Introduction The Benjamin–Feir type modulational instability [1,2] is one of hallmarks of nonlinear waves [3] described by usual nonlinear Schr¨ odinger equations (NLSE), where the cubic nonlinear term plays the role. The NLSE has received substantial interests from a wide discipline such as water waves, plasma waves, nonlinear optics, fluid dynamics, partial differential equations, quantum mechanics and many others [4]. Recently, a new logarithmic nonlinear Schr¨ odinger equation (NLSE) was proposed for the wave packet dynamics under continuous quantum measurement [5,6]. Therefore, one important question is how the modulational instability (MI) is formulated for this logarithmic NLSE and this is answered by this Letter. Formally, replacing the time derivative in the linear Schr¨odinger equation as ∂/∂t → ∂/∂t+κA(ψ)+Γ B(ψ) with coupling strengths κ ∈ R and Γ ∈ R makes various nonlinear Schr¨odinger equations: E-mail address: [email protected]. 13914485239. 1 Tel.: +86 ∂ i~ + κA(ψ) + Γ B(ψ) ψ(x, t) = Hψ(x, t), (x, t ∈ R) (1) ∂t http://dx.doi.org/10.1016/j.laa.2014.09.031 by Elsevier Inc. with the nonlinear 0024-3795/© terms A(ψ)2014 andPublished B(ψ), respectively. The linear Schr¨odinger equation (κ = Γ = 0), the usual cubic NLSE (κ ̸= 0, Γ = 0, A(ψ) = −i|ψ|2 ), the cubic–quintic NLSE (κ ̸= 0, Γ ̸= 0, A(ψ) = −i|ψ|2 , E-mail address: [email protected]. http://dx.doi.org/10.1016/j.aml.2015.03.020 0893-9659/© 2015 Elsevier Ltd. All rights reserved.
T. Yamano / Applied Mathematics Letters 48 (2015) 124–127
125
B(ψ) = −i|ψ|4 ), and the logarithmic NLSE (κ ̸= 0, Γ = 0, A(ψ) = ln |ψ|2 − ⟨ln |ψ|2 ⟩) [5] are included in this expression as special cases. In this work, we consider the extended logarithmic NLSE in one-dimensional 2 2 ~ d ) [6]: case for the free particle Hamiltonian of mass m (i.e., H = − 2m dx2 Γ ∂ ψ(x, t) ψ(x, t) 2 2 i~ + κ ln |ψ(x, t)| − ⟨ln |ψ(x, t)| ⟩ − ln − ln ψ(x, t) = Hψ(x, t) (2) ¯ t) ¯ t) ∂t 2 ψ(x, ψ(x, where the brackets ⟨⟩ denote the mean with |ψ(x, t)|2 over the entire domain of the position variable x and ¯ t) the complex conjugate. This equation admits the wave packet dynamics within the Bohmian quantum ψ(x, trajectories formalism [7,8] and in conformity with the theory of continuous quantum measurements [9]. The nonlinear coefficients κ and Γ are assumed to be positive, and have interpretations of the resolution of the measurement and the friction associated with the quantum dissipative dynamics, respectively [6]. A stability of the width of the Gaussian wave packet as a solution of this NLSE is discussed in Ref. [10]. 2. Modulational instability MI is a significant property as it relates to a way of collapse of a plane wave. A recent historical review on MI is given in Ref. [11] and its standard treatment for the cubic NLSE with envelope soliton solutions is well articulated (e.g. [12,13]). First, in this section, to obtain an analog of the dispersion relation, we insert the plane wave form ψ(x, t) = ψ0 eiS(x,t)/~ into Eq. (2) with a real constant amplitude ψ0 and the phase S(x, t), which is specified later. This gives ψ0 2 i ~2 − 2 Sx + ψ0 Sxx − Γ (S − ⟨S⟩) ψ0 . (3) − S t ψ0 = − 2m ~ ~ Thus, we have 1 (i~Sxx − Sx2 ) + Γ (S − ⟨S⟩). (4) 2m Next, introducing the perturbation in amplitude and phase in the plane wave form as ψ(x, t) = (ψ0 + ϵ(x, t))ei(S(x,t)+θ(x,t))/~ , where ϵ(x, t) and θ(x, t) are treated as real functions at this stage [13], and substituting these into Eq. (2) gives i i~ ϵt + (ψ0 + ϵ)(St + θt ) + i2~κ [ln(ψ0 + ϵ) − ⟨ln(ψ0 + ϵ)⟩] (ψ0 + ϵ) + Γ [(S − ⟨S⟩) + (θ − ⟨θ⟩)] (ψ0 + ϵ) ~ i 1 ~2 =− ϵxx − 2 (ψ0 + ϵ)(Sx + θx )2 + {2ϵx (Sx + θx ) + (ψ0 + ϵ)(Sxx + θxx )} . (5) 2m ~ ~ St =
Assuming the perturbations being carrier wave forms with a wave number δ and an angular frequency ν, i.e., ϵ(x, t) = ϵ0 ei(δx−νt) and θ(x, t) = θ0 ei(δx−νt) , where ϵ0 and θ0 are the real amplitudes. The real and the imaginary parts of the above equation read respectively as 1 ~2 [(St − iνθ) − Γ (S − ⟨S⟩ + θ − ⟨θ⟩)] (ψ0 + ϵ) = − δ 2 ϵ + 2 (ψ0 + ϵ)(Sx + iδθ)2 (6) 2m ~ and − iν~ϵ + 2~κ [ln(ψ0 + ϵ) − ⟨ln(ψ0 + ϵ)⟩] (ψ0 + ϵ) = −
~ 2iδϵ(Sx + iδθ) + (ψ0 + ϵ)(Sxx − δ 2 θ) . 2m
(7)
We linearize both equations by dropping terms ϵθ and θ2 . Noting ⟨ϵ⟩ = ⟨θ⟩ = 0 and ln(ψ0 + ϵ) ∼ ln ψ0 + ϵ/ψ0 for a small perturbation (i.e., ϵ0 ≪ ψ0 ), Eqs. (6) and (7) respectively reduce to 1 2 2 δSx 1 2 St − Γ (S − ⟨S⟩) + (~ δ + Sx2 ) ϵ + i − ν − Γ ψ0 θ = − St − Γ (S − ⟨S⟩) + Sx ψ0 (8) 2m m 2m
126
T. Yamano / Applied Mathematics Letters 48 (2015) 124–127
2 ~ Fig. 1. Plot of the growth rate as a function of the scaled wave number by ( 2m Γ) .
and
Sxx ~ + 2κ + i 2m
δSx −ν m
ϵ−
~δ 2 ~ ψ0 θ = − ψ0 Sxx . 2m 2m
Using the relation Eq. (4), Eq. (8) is further simplified as δSx ψ0 ~ (iSxx + ~δ 2 ) ϵ + i − ν − Γ ψ0 θ = −i Sxx . 2m m 2m
(9)
(10)
Our particular interest for investigating MI is realized when the phase is fixed as S(x, t) = ~(kx − Ω t) with a positive wave number k and an angular frequency Ω . In this case, the dispersion relation becomes 2 Ω = ~k 2m − Γ k(x − ⟨x⟩). Therefore, the simultaneous equation regarding ϵ and θ to consider is 2 2 δ ~k ~ ϵ+ i δ − ν − Γ ψ0 θ = 0 2m m (11) ~k δ2 i δ − ν + 2κ ϵ − ψ0 θ = 0. m 2m Hence, the non-trivial solution is feasible when the determinant of the matrix of coefficients vanishes: 2 2 ~δ 2 ~k ~k − + 2κΓ + δ−ν δ − ν ψ0 = 0 (12) ψ0 − i(2κ − Γ ) 2m m m which indicates Γ = 2κ and leads to 2 ~k ~δ ν= δ± − Γ 2. (13) m 2m ~δ 2 Therefore, the MI is possible when 2m < Γ 2 , since ν becomes a complex number, indicating the ex ~ 2 ponential growth of the perturbation. Furthermore, the growth rate Γ 1 − 2mΓ δ 2 shown in Fig. 1 approximately equals to the friction coefficient Γ in the long wave limit of the perturbation (δ 2 ≈ 0), which constitutes another interpretation for the nonlinear coefficient contained in the present logarithmic NLSE. 3. Remarks and discussions The result reported here contrasts with the case of the cubic NLSE, iψt + pψxx+ q|ψ|2 ψ = 0, where MI occurs when the coefficients satisfy pq > 0 with the growth rate Im(ν) = pψ0 δ 2q/p − (δ/ψ0 )2 . The
T. Yamano / Applied Mathematics Letters 48 (2015) 124–127
127
2 curve starts increasing from zero at δ/ψ = 0 towards a peak value qψ at δ/ψ = q/p, then decreasing 0 0 0 towards zero at δ/ψ0 = 2q/p. The additional higher order nonlinear terms to the cubic NLSE such as double power nonlinearities alter only the equation that comes from the real part and admits the similar form of the growth rate. Indeed, for example, the cubic–quartic NLSE iψt + pψxx + q|ψ|2 ψ + r|ψ|3 ψ = 0 has 2q MI when r < 3ψ0 with its growth rate Im(ν) = pψ0 δ (2q + 3rψ0 )/p − (δ/ψ0 )2 [14]. This indicates that the addition of power nonlinearity is an effective mean to shift the peak of the growth rate and the modulational wave number at which it vanishes towards larger values by 3rψ0 . On the other hand, the present logarithmic NLSE shows different MI feature; the maximum in the growth rate is attained at vanishing modulational wave number, meaning that the instability increases far rapidly at lower frequencies of the perturbation. This situation is reminiscent of the gain spectrum of the polarization instability of a light pulse propagating in a nonlinear birefringent dispersive medium such as optical fibers [15]. For the pulses in fiber, a coherently coupled NLSE is employed to show the behavior of gain curve by means of the standard MI analysis. The associated gain as a function of the input power is peaked at zero modulational frequency and decreases to zero just as the curve shown in Fig. 1. Therefore, this study demonstrated also that the different class of NLSEs shows the similar behavior in MI growth curves. Acknowledgments The author thanks researchers at Tokyo City University for extensive discussions in 2008 on the cubic–quartic NLSE, whereby his interest was roused on this topic. References [1] T.B. Benjamin, J.E. Feir, The disintegration of wave trains on deep water, Part 1. Theory, J. Fluid Mech. 27 (1967) 417. [2] T.B. Benjamin, Instability of periodic wavetrains in nonlinear dispersive systems, Proc. R. Soc. Lond. Ser. A 299 (1967) 59. [3] G.B. Whitham, Linear and Nonlinear Waves, in: Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts, Wiley-Interscience, 1999. [4] C. Sulem, P.-L. Sulem, The Nonlinear Schr¨ odinger Equation: Self-Focusing and Wave Collapse, in: Applied Mathematical Sciences, vol. 139, Springer, 1999, Section 1.3. [5] A.B. Nassar, Quantum trajectories and the Bohm time constant, Ann. Phys. 331 (2013) 317. [6] A.B. Nassar, S. Miret-Art´ es, Dividing line between quantum and classical trajectories in a measurement problems: Bohmian time constant, Phys. Rev. Lett. 111 (2013) 15041. [7] D. Bohm, A suggested interpretation of the quantum theory in terms of “hidden” variables I, Phys. Rev. 85 (1952) 166. [8] D. Bohm, A suggested interpretation of the quantum theory in terms of “hidden” variables II, Phys. Rev. 85 (1952) 180. [9] M.B. Mensky, Continuous Quantum Measurement and Path Integrals, IOP Publisher, Bristol, 1993. [10] C. Zander, A.R. Plastino, J. D´ıaz-Alonso, Wave packet dynamics for a nonlinear Schr¨ odinger equation describing continuous position measurements, 2015), preprint. [11] V.E. Zakharov, L.A. Ostrovsky, Physica D 238 (2009) 540. [12] T. Dauxois, M. Peyrard, Physics of Solitons, Cambridge University Press, 2006. [13] S. Watanabe, Introduction to Soliton Physics, Baifukan, 1985, Section 5.5 (in Japanese). [14] T. Yamano, Phase portrait analysis for the envelope solution of cubic-quartic nonlinear Schr¨ odinger equations and modulational instability, Hokkaido University Technical Report Series in Mathematics (ISSN 1348-4338) #140, 2009, pp. 51–55 (in Japanese). [15] S. Wabnitz, Modulational polarization instability of light in a nonlinear birefringent dispersive medium, Phys. Rev. A 38 (1988) 2018.
Contents lists at ScienceDirect Contents lists available at available ScienceDirect
LinearMathematics Algebra andLetters its Applications Applied www.elsevier.com/locate/laa www.elsevier.com/locate/aml
eigenvalue problem of Jacobi matrix Modulational Inverse instability for a logarithmic nonlinear Schr¨odinger with mixed data equation Takuya Yamano Ying Wei 1 Department of Mathematics and Physics, Faculty of Nanjing Science,University Kanagawa of University, 2946, Tsuchiya, Hiratsuka, Department of Mathematics, Aeronautics and6-233 Astronautics, Kanagawa 259-1293, Japan Nanjing 210016, PR China
article
i n faor t i c l e
ianbfs ot r a c t
a b s t r a c t
Article history: Article history: In this paper, the inverse problem of reconstructing We investigate the modulational instability for eigenvalue a logarithmic nonlinear Schr¨ odinger 2014 Received 20 March 2015Received 16 January equation a Jacobi matrix infrom eigenvalues, its leading principal that has recently proposed the its context of a wave packet dynamics Accepted 20 September 2014 Received in revised form 30 March submatrix and part process. of the eigenvalues of itslimit submatrix under a continuous quantum measurement In the long wave of the Available online 22 October 2014 2015 considered. Therate necessary and sufficient conditions for perturbation, we showis that the growth of the instability coincides with the Submitted by Y. Wei Accepted 31 March 2015 the existence and uniqueness of the solution are derived. friction coefficient contained in the equation. Available online 20 April 2015 Furthermore, a numerical algorithm and © 2015 Elsevier Ltd. Allsome rightsnumerical reserved. MSC: examples are given. 15A18 Keywords: © 2014 Published by Elsevier Inc. Modulational instability15A57 Logarithmic nonlinear Schr¨ odinger Keywords: equation Jacobi matrix Plane wave Eigenvalue Inverse problem Submatrix
1. Introduction The Benjamin–Feir type modulational instability [1,2] is one of hallmarks of nonlinear waves [3] described by usual nonlinear Schr¨ odinger equations (NLSE), where the cubic nonlinear term plays the role. The NLSE has received substantial interests from a wide discipline such as water waves, plasma waves, nonlinear optics, fluid dynamics, partial differential equations, quantum mechanics and many others [4]. Recently, a new logarithmic nonlinear Schr¨ odinger equation (NLSE) was proposed for the wave packet dynamics under continuous quantum measurement [5,6]. Therefore, one important question is how the modulational instability (MI) is formulated for this logarithmic NLSE and this is answered by this Letter. Formally, replacing the time derivative in the linear Schr¨odinger equation as ∂/∂t → ∂/∂t+κA(ψ)+Γ B(ψ) with coupling strengths κ ∈ R and Γ ∈ R makes various nonlinear Schr¨odinger equations: E-mail address: [email protected]. 13914485239. 1 Tel.: +86 ∂ i~ + κA(ψ) + Γ B(ψ) ψ(x, t) = Hψ(x, t), (x, t ∈ R) (1) ∂t http://dx.doi.org/10.1016/j.laa.2014.09.031 by Elsevier Inc. with the nonlinear 0024-3795/© terms A(ψ)2014 andPublished B(ψ), respectively. The linear Schr¨odinger equation (κ = Γ = 0), the usual cubic NLSE (κ ̸= 0, Γ = 0, A(ψ) = −i|ψ|2 ), the cubic–quintic NLSE (κ ̸= 0, Γ ̸= 0, A(ψ) = −i|ψ|2 , E-mail address: [email protected]. http://dx.doi.org/10.1016/j.aml.2015.03.020 0893-9659/© 2015 Elsevier Ltd. All rights reserved.
T. Yamano / Applied Mathematics Letters 48 (2015) 124–127
125
B(ψ) = −i|ψ|4 ), and the logarithmic NLSE (κ ̸= 0, Γ = 0, A(ψ) = ln |ψ|2 − ⟨ln |ψ|2 ⟩) [5] are included in this expression as special cases. In this work, we consider the extended logarithmic NLSE in one-dimensional 2 2 ~ d ) [6]: case for the free particle Hamiltonian of mass m (i.e., H = − 2m dx2 Γ ∂ ψ(x, t) ψ(x, t) 2 2 i~ + κ ln |ψ(x, t)| − ⟨ln |ψ(x, t)| ⟩ − ln − ln ψ(x, t) = Hψ(x, t) (2) ¯ t) ¯ t) ∂t 2 ψ(x, ψ(x, where the brackets ⟨⟩ denote the mean with |ψ(x, t)|2 over the entire domain of the position variable x and ¯ t) the complex conjugate. This equation admits the wave packet dynamics within the Bohmian quantum ψ(x, trajectories formalism [7,8] and in conformity with the theory of continuous quantum measurements [9]. The nonlinear coefficients κ and Γ are assumed to be positive, and have interpretations of the resolution of the measurement and the friction associated with the quantum dissipative dynamics, respectively [6]. A stability of the width of the Gaussian wave packet as a solution of this NLSE is discussed in Ref. [10]. 2. Modulational instability MI is a significant property as it relates to a way of collapse of a plane wave. A recent historical review on MI is given in Ref. [11] and its standard treatment for the cubic NLSE with envelope soliton solutions is well articulated (e.g. [12,13]). First, in this section, to obtain an analog of the dispersion relation, we insert the plane wave form ψ(x, t) = ψ0 eiS(x,t)/~ into Eq. (2) with a real constant amplitude ψ0 and the phase S(x, t), which is specified later. This gives ψ0 2 i ~2 − 2 Sx + ψ0 Sxx − Γ (S − ⟨S⟩) ψ0 . (3) − S t ψ0 = − 2m ~ ~ Thus, we have 1 (i~Sxx − Sx2 ) + Γ (S − ⟨S⟩). (4) 2m Next, introducing the perturbation in amplitude and phase in the plane wave form as ψ(x, t) = (ψ0 + ϵ(x, t))ei(S(x,t)+θ(x,t))/~ , where ϵ(x, t) and θ(x, t) are treated as real functions at this stage [13], and substituting these into Eq. (2) gives i i~ ϵt + (ψ0 + ϵ)(St + θt ) + i2~κ [ln(ψ0 + ϵ) − ⟨ln(ψ0 + ϵ)⟩] (ψ0 + ϵ) + Γ [(S − ⟨S⟩) + (θ − ⟨θ⟩)] (ψ0 + ϵ) ~ i 1 ~2 =− ϵxx − 2 (ψ0 + ϵ)(Sx + θx )2 + {2ϵx (Sx + θx ) + (ψ0 + ϵ)(Sxx + θxx )} . (5) 2m ~ ~ St =
Assuming the perturbations being carrier wave forms with a wave number δ and an angular frequency ν, i.e., ϵ(x, t) = ϵ0 ei(δx−νt) and θ(x, t) = θ0 ei(δx−νt) , where ϵ0 and θ0 are the real amplitudes. The real and the imaginary parts of the above equation read respectively as 1 ~2 [(St − iνθ) − Γ (S − ⟨S⟩ + θ − ⟨θ⟩)] (ψ0 + ϵ) = − δ 2 ϵ + 2 (ψ0 + ϵ)(Sx + iδθ)2 (6) 2m ~ and − iν~ϵ + 2~κ [ln(ψ0 + ϵ) − ⟨ln(ψ0 + ϵ)⟩] (ψ0 + ϵ) = −
~ 2iδϵ(Sx + iδθ) + (ψ0 + ϵ)(Sxx − δ 2 θ) . 2m
(7)
We linearize both equations by dropping terms ϵθ and θ2 . Noting ⟨ϵ⟩ = ⟨θ⟩ = 0 and ln(ψ0 + ϵ) ∼ ln ψ0 + ϵ/ψ0 for a small perturbation (i.e., ϵ0 ≪ ψ0 ), Eqs. (6) and (7) respectively reduce to 1 2 2 δSx 1 2 St − Γ (S − ⟨S⟩) + (~ δ + Sx2 ) ϵ + i − ν − Γ ψ0 θ = − St − Γ (S − ⟨S⟩) + Sx ψ0 (8) 2m m 2m
126
T. Yamano / Applied Mathematics Letters 48 (2015) 124–127
2 ~ Fig. 1. Plot of the growth rate as a function of the scaled wave number by ( 2m Γ) .
and
Sxx ~ + 2κ + i 2m
δSx −ν m
ϵ−
~δ 2 ~ ψ0 θ = − ψ0 Sxx . 2m 2m
Using the relation Eq. (4), Eq. (8) is further simplified as δSx ψ0 ~ (iSxx + ~δ 2 ) ϵ + i − ν − Γ ψ0 θ = −i Sxx . 2m m 2m
(9)
(10)
Our particular interest for investigating MI is realized when the phase is fixed as S(x, t) = ~(kx − Ω t) with a positive wave number k and an angular frequency Ω . In this case, the dispersion relation becomes 2 Ω = ~k 2m − Γ k(x − ⟨x⟩). Therefore, the simultaneous equation regarding ϵ and θ to consider is 2 2 δ ~k ~ ϵ+ i δ − ν − Γ ψ0 θ = 0 2m m (11) ~k δ2 i δ − ν + 2κ ϵ − ψ0 θ = 0. m 2m Hence, the non-trivial solution is feasible when the determinant of the matrix of coefficients vanishes: 2 2 ~δ 2 ~k ~k − + 2κΓ + δ−ν δ − ν ψ0 = 0 (12) ψ0 − i(2κ − Γ ) 2m m m which indicates Γ = 2κ and leads to 2 ~k ~δ ν= δ± − Γ 2. (13) m 2m ~δ 2 Therefore, the MI is possible when 2m < Γ 2 , since ν becomes a complex number, indicating the ex ~ 2 ponential growth of the perturbation. Furthermore, the growth rate Γ 1 − 2mΓ δ 2 shown in Fig. 1 approximately equals to the friction coefficient Γ in the long wave limit of the perturbation (δ 2 ≈ 0), which constitutes another interpretation for the nonlinear coefficient contained in the present logarithmic NLSE. 3. Remarks and discussions The result reported here contrasts with the case of the cubic NLSE, iψt + pψxx+ q|ψ|2 ψ = 0, where MI occurs when the coefficients satisfy pq > 0 with the growth rate Im(ν) = pψ0 δ 2q/p − (δ/ψ0 )2 . The
T. Yamano / Applied Mathematics Letters 48 (2015) 124–127
127
2 curve starts increasing from zero at δ/ψ = 0 towards a peak value qψ at δ/ψ = q/p, then decreasing 0 0 0 towards zero at δ/ψ0 = 2q/p. The additional higher order nonlinear terms to the cubic NLSE such as double power nonlinearities alter only the equation that comes from the real part and admits the similar form of the growth rate. Indeed, for example, the cubic–quartic NLSE iψt + pψxx + q|ψ|2 ψ + r|ψ|3 ψ = 0 has 2q MI when r < 3ψ0 with its growth rate Im(ν) = pψ0 δ (2q + 3rψ0 )/p − (δ/ψ0 )2 [14]. This indicates that the addition of power nonlinearity is an effective mean to shift the peak of the growth rate and the modulational wave number at which it vanishes towards larger values by 3rψ0 . On the other hand, the present logarithmic NLSE shows different MI feature; the maximum in the growth rate is attained at vanishing modulational wave number, meaning that the instability increases far rapidly at lower frequencies of the perturbation. This situation is reminiscent of the gain spectrum of the polarization instability of a light pulse propagating in a nonlinear birefringent dispersive medium such as optical fibers [15]. For the pulses in fiber, a coherently coupled NLSE is employed to show the behavior of gain curve by means of the standard MI analysis. The associated gain as a function of the input power is peaked at zero modulational frequency and decreases to zero just as the curve shown in Fig. 1. Therefore, this study demonstrated also that the different class of NLSEs shows the similar behavior in MI growth curves. Acknowledgments The author thanks researchers at Tokyo City University for extensive discussions in 2008 on the cubic–quartic NLSE, whereby his interest was roused on this topic. References [1] T.B. Benjamin, J.E. Feir, The disintegration of wave trains on deep water, Part 1. Theory, J. Fluid Mech. 27 (1967) 417. [2] T.B. Benjamin, Instability of periodic wavetrains in nonlinear dispersive systems, Proc. R. Soc. Lond. Ser. A 299 (1967) 59. [3] G.B. Whitham, Linear and Nonlinear Waves, in: Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts, Wiley-Interscience, 1999. [4] C. Sulem, P.-L. Sulem, The Nonlinear Schr¨ odinger Equation: Self-Focusing and Wave Collapse, in: Applied Mathematical Sciences, vol. 139, Springer, 1999, Section 1.3. [5] A.B. Nassar, Quantum trajectories and the Bohm time constant, Ann. Phys. 331 (2013) 317. [6] A.B. Nassar, S. Miret-Art´ es, Dividing line between quantum and classical trajectories in a measurement problems: Bohmian time constant, Phys. Rev. Lett. 111 (2013) 15041. [7] D. Bohm, A suggested interpretation of the quantum theory in terms of “hidden” variables I, Phys. Rev. 85 (1952) 166. [8] D. Bohm, A suggested interpretation of the quantum theory in terms of “hidden” variables II, Phys. Rev. 85 (1952) 180. [9] M.B. Mensky, Continuous Quantum Measurement and Path Integrals, IOP Publisher, Bristol, 1993. [10] C. Zander, A.R. Plastino, J. D´ıaz-Alonso, Wave packet dynamics for a nonlinear Schr¨ odinger equation describing continuous position measurements, 2015), preprint. [11] V.E. Zakharov, L.A. Ostrovsky, Physica D 238 (2009) 540. [12] T. Dauxois, M. Peyrard, Physics of Solitons, Cambridge University Press, 2006. [13] S. Watanabe, Introduction to Soliton Physics, Baifukan, 1985, Section 5.5 (in Japanese). [14] T. Yamano, Phase portrait analysis for the envelope solution of cubic-quartic nonlinear Schr¨ odinger equations and modulational instability, Hokkaido University Technical Report Series in Mathematics (ISSN 1348-4338) #140, 2009, pp. 51–55 (in Japanese). [15] S. Wabnitz, Modulational polarization instability of light in a nonlinear birefringent dispersive medium, Phys. Rev. A 38 (1988) 2018.